How to control complete model with single empty
do you mean like this?
https://youtu.be/k_ks-DzSBV0
if yes:
Parenting the empty works exactly as you described.
But...
you have to change your limit rotation settings.
I just had a look at your fold constraint.
The correct setup looks like this:

You have to restrict all axis except the one you want to move (in your special case here) so just the x-Axis got some values, all others with 0 but checked.
I am not sure why it does not work if you check the lock on the axis values, but with my settings it works.